Talk:Number

But what _is_ a number?

"... an abstract idea used in counting and measuring" has no value as a definition, especially as the concept is not further visited. There are plenty of abstract ideas used in counting and measuring -- operators, equality, order -- the "definition" seems to hope that the reader already knows a number when she sees it.

I would propose the following definition: a numeric system is a monoid whose operator preserves a prescribed total order, and a number is an element of a numeric system, or more specifically one of the canonical numeric systems elevated by mathematicians over the centuries. This highlights the essential interface that we have had with numbers from pre-history: we can add them to each other and we can compare them against one another. Of course, in many numeric systems we can do more, but it seems the core that places numbers so fundamentally at the heart of our understanding of the world.

The "flaw" with this definition, of course, is that it excludes the complex field which has no inherent total order. I'm not willing to undertake WP:BOLD without feedback because it may be a heretical notion, but I don't believe this to actually be a flaw: I don't see that the elements of a complex field are much different than a matrix of numbers or a polynomial: an extension of a numeric system that uses the underlying numbers to form rich algebraic structures.

So shall I take a stab at editing the page, or is this a non-starter and people actually like numbers being an abstract idea used in counting and measuring? MatthewDaly 02:44, 6 November 2007 (UTC)

I'm afraid we're not permitted to make up our own definitions in WP articles. Doesn't matter whether they're good or not, so I won't address your proposal on the merits. Please review WP:NOR. --Trovatore 03:04, 6 November 2007 (UTC)

I am unclear on the scope of your rejection. The introduction is completely unsourced, so someone seems to have made up the "abstract idea" definition. It is hardly original research to observe that the concept of numbers historically have been about computability and order, as these are the whole of the core of numerical structures of Peano, Dedekind, and Conway, whose work I would intend to both leverage and reference were I to help on this page. I can appreciate that there is a lack of unanimity among mathematicians when it comes to the "numberness" of mathematical structures that are generated from the real numbers but have increasingly pathological behavior (complex numbers and polynomials are not ordered, matrices and quaternions don't have commutative multiplications, etc.), but the current alternative of devising a definition that is so vague as to not clearly indicate things that are universally regarded as not numbers strikes me as a poor one.MatthewDaly 05:14, 6 November 2007 (UTC)

Well, I was indeed rejecting your proposal (as it relates to the article), but I was not specifically defending the current text (which actually I hadn't read recently). There is no accepted general definition of "number", and the article certainly should not give the impression that there is.

However, now that I take a brief glance at the existing text, I don't see anything terribly wrong with it. I see the assertion "[a] number is an abstract idea used in counting and measuring" as being a descriptive assertion rather than a definition, and one that should be pretty uncontroversial -- that is, we all agree that the abstract idea of number is used in counting and measuring, whether or not we think that this usage precisely isolates what it means to be a number. Maybe you'd like to propose some text that makes more explicit that the sentence is not a definition?

Now that leaves open the question of whether the article should discuss definitions (necessarily plural, I think) that have been proposed for the notion of "number" in general. I don't think it's terribly necessary -- and I certainly wouldn't put it in the lead section -- but it might make an interesting sidelight somewhere in the body of the article. But any such definitions need to be sourced, and it should not be implied that any of them has general acceptance, because I think that none of them does. --Trovatore 06:35, 6 November 2007 (UTC)

Has MatthewDaly been reading Mathematics Made Difficult (ISBN 0-7234-0415-1)? That definition looks as if it was from there. — Arthur Rubin | (talk) 14:24, 6 November 2007 (UTC)

I have not, I'll have to look it up. It seems to me exactly the definition that any formalist would devise; I wonder why they do not seem to have made a point of doing so. Perhaps it doesn't have the same utility as we got from axiomatizing the previously abstract notions of "set" and "proof", but at least it would allow students to understand why some mathematical objects are universally understood to be numbers while others are universally excluded. Ah well, I am disappointed but sanguine.MatthewDaly 17:41, 7 November 2007 (UTC)

A number system doesn't have to be a monoid. If you define a number system to be a type of monoid, then the Octonions are not a number system, as they are nonassociative. Willow1729 (talk) 23:31, 21 September 2008 (UTC)

In Euclidean (plane) Geometry a straight line always has a large number of point locations. Then, if you buy a location point of the number zero and the number 1, you have a method of creating an ordered set of integer numbered distances on the line, which can increase up to but not include a value for infinity. Then there are other categories of distances on the number line, like the rational number (m/n) values, and the square root of diagonals values as developed by the Pythagorean theorem. And since these square root values have real distance locations on the number line, they are still considered to be real numbers, but as having an irrational name. And finally there are number names like Pi that dont have a determinal point location on the straight number line, and also point locations with neither a name nor a determinal location. This is all part of the rudiments of mathematical development whuch most people have forgotten or never been exposed to. WFPM (talk) 20:48, 9 November 2008 (UTC)

I donr think that the number Pi can be considered to be a real number since it's exact value connot be located on the number line.WFPM (talk) 21:59, 9 November 2008 (UTC)

Whether this is a suggestion as to improvement of the article rather than commentaty about the subject (which should be summarily deleted), it's nonsense. Not even intuitionists and constructivists believe that π is not a real number. And finitists don't believe there is such a thing as a real number.... — Arthur Rubin (talk) 22:15, 9 November 2008 (UTC)

So now, in addition to all the distances on the number line, you want to creare a category of real numbers whose exact location on the number line cannot be located. However I.ve got to admit that you could come pretty close by rolling a circle as per the article. But that' s not plane geometry as defined by Euclid. But a Point on the number line isn't defined by an approximate location, and I thought that real numbers were. But I'm not a nit picker, just a definition picker. WFPM (talk) 00:42, 10 November 2008 (UTC)

And I'm not an institutionist or a constructivist or a finitist. Im just an Engineer interested in science.WFPM (talk) 00:47, 10 November 2008 (UTC)

PS Following Mathematical logic, we can agree that if we knew the real value of Pi it would be a real number. But since we only have an approximate albeit pretty accurate estimate, we cant say that our estimate represents the real value of Pi. WFPM (talk) 00:53, 10 November 2008 (UTC)

←
I would say that any attempt to define a number would be OR. Numbers are those things we have decided to call numbers. We have good definitions of particular types of numbers: Natural numbers, real numbers etc. but not of the word number, which is more a lingustic device rather than a mathematical definition. --Salix (talk): 01:50, 10 November 2008 (UTC)

Yes but besides a location of a point or a magnitude of a number, we have the definition of things. And a point doesn't have a dimension, just a unique location. And a number doesn't have any other significance other than being the distance involved with a point on the ordered number line. And if you cant find the point, the number becomes indeterminate. WFPM (talk) 02:28, 10 November 2008 (UTC)

If it's just your opinion, even if justified, it has no place in a Wikipedia article, unless you can find some reliable source which reports that π is not a real number. Please see original research. I've never seen anything like that in print or on the web. — Arthur Rubin (talk) 07:49, 10 November 2008 (UTC)

I guess you're right. The real number Pi would be a real number. But what do you do if you cant find it's location on the number line? And where else are you going to find out it's true value? And I use the Pi and epsilon values often because they're in my Casio, and I need them as mathematical tools for approximate calculations. And I even call up a random number occasionally, even though I know that it is not really a random number. But I would think that if you are going to have an article about numbers you would start out by defining what a number really is. And the only think that I can think of that it really is is a distance from zero to a point on the number line.WFPM (talk) 15:46, 10 November 2008 (UTC)

I notice that Wikipedia also has an article on Ratio, which also involves mathematical quantities. And I'm trying to think of a ratio that has a quantity that couldn't be determined as being a real number and thus a point on the number line. But I cant think of any.WFPM (talk) 16:31, 10 November 2008 (UTC)

To find pi on the number line, make a circular disk whose diameter is equal to the distance from zero to one on the number line. Make a mark on the edge of the disk and put that point on the disk on zero. Roll the disk to the right, without slipping, and when the mark comes down and touches the number line, that point is pi.

A number can have a measurement attached, 12 eggs or 4 meters. The measurement is called a "dimension". A ratio is a number without a measurement. Pi is a ratio -- whether we measure a circle using inches or meters, it doesn't matter, pi always comes out the same. Ratios are calculated using division. Pi, for example, is circumfrance divided by diameter. Sometimes raitos are written as fractions, 5/4, and sometimes with a colon 5:4 (read the ratio of five to four. The rule for equal ratios is, two ratios are equal if the product of their means equals the product of their extremes. Thus a:b = c:d if and only if bc = ad.Rick Norwood (talk) 18:33, 10 November 2008 (UTC)

All numbers are number without measurement. Measurements are just mathematical applications of the set of numbers. Except for a conceived ordered number line, which is supposed to include all numbers together with their appropriate distance locations from zero on the number line. And I'll agree that Pi is a ratio quantity, since you can't determine it mathematically. But the number line should be considered as being made up of an infinity of points, and therefore infinitely divisable. But all of the points are not quantitatively determinable. In fact it is said that there are an infinity of numbers on the number line within the range of zero to 1. WFPM (talk) 19:02, 10 November 2008 (UTC)

And I agree with you that Euclid should have been smart enough to have used a circular piece of paper to measure out Pi distance on the number line.WFPM (talk) 03:29, 19 November 2008 (UTC)

Ok, I'll stick my neck out and claim that π can be determined mathematically. By the Leibniz formula for pi it is four times 1/1 - 1/3 + 1/5 - 1/7 + ... Given any rational number, finitely many terms of this sequence suffice to decide whether that number is smaller or larger than π (it can never be equal to π because π is irrational). Therefore π cuts the rational line at a well-defined point, and this according to Dedekind is what it means to be a well-defined real number. If you don't agree your quarrel is with Dedekind, not me. --Vaughan Pratt (talk) 19:48, 3 January 2009 (UTC)

A number is

A number is a word(concept) that represents and contains a sequence of patterns or data. One word, equals one object.

For instance the numeral one, represents the mental object of one, as well as the geometric-symbol of one.

The numeral one could be considered the first letter of the mathematical alphabet. I think it's best to think of mathematics as a language that describes shapes and patterns. A number would be considered both data and a shape at the same time, a data-shape. In fact all numerals are geometric shapes. -- Yours truly BeExcellent2every1 (talk) 12:24, 21 November 2007 (UTC)

This is original research and not suitable for Wikipedia. Rick Norwood (talk) 15:30, 19 November 2007 (UTC)

Catalan deffinition of number

I suggest the definition given by Pompeu Fabra in his Dictionary of Catalan Language.

It can be translated into English like:

A number is the concept that arises from counting things which form a collection, or a generalization of this concept.

This definition has several advantages:

• It is clear for everybody
• Directly relates numbers with intuitive groundings of set theory (collection).
• Includes all kinds of numbers, because all of them can be considered generalizations of natural numbers (those which arise from counting).
• Excludes all things that are not considered numbers.
• It is not original research. It is the definition given by an expert both on mathematics and on language. There is a clear bibliographic reference.

I don’t know if it is the best solution for the English Wikipedia. In Catalan there are three completely different words to express: a) "nombre", the abstract concept of number (the definition I am suggesting), b) "número", the representation of the number in a numeration system, and c) "xifra" the symbols used to represent the numbers. I fear this is not the same in English, but from a mathematical point of view when we talk about numbers we think on the abstract concept. --62.57.139.143 (talk) 12:27, 1 January 2008 (UTC)

It seems very similar to the current opening sentence, except that we refer to measurement as well as counting. I think this is a good thing, as measurement is significant in its own right, not just a generalisation of counting. JPD (talk) 11:25, 2 January 2008 (UTC)

As it has been said before: “There are plenty of abstract ideas used in counting and measuring”.

A lot of confusion comes from involving the measuring process in defining number.

Measuring can be reduced to counting how many times the units of measure are contained in the magnitude to be measured.

But in the process of measuring, several problems have to be solved:

• How the units of measurement are combined to generate a higher magnitude? i.e. the units of length have to be put contiguously to other units of length on a straight line, it is forbidden to overlap, left gaps and put them on a curved line.
• How to compare two magnitudes? You have to describe the experiment used to compare the weight to be measured against the units of measure using a mechanical device.
• How to divide a magnitude in equally valued parts? How to divide the unit of measure to get factionary units?

They are mainly related to the physic properties of the magnitude to be measured.

Measuring can be related to the invention of rational numbers. This is because of the need of dividing the measuring unit in equally valued parts. This could be avoided if the unit of measure is small enough. It also can be related to invention of real numbers if it is admitted that magnitude can be divided in infinitesimally small parts (which is not clear from practical and even theoretical point of view when considering real magnitudes). But they can be introduced simply as generalizations of natural numbers.

But, the only new concept that arises from counting is the concept of natural number. Counting is establishing a bijection between the set being counted and a set of new entities, the set of natural numbers, the only meaning of its elements is that that have in common all the sets that give the same outcome when being counted. That’s why I think it is better to use “The concept that arises from counting” instead of “an abstract idea used in counting”.

I think that making reference to measuring has the intention of opening doors to generalizations of natural numbers, but there are other kinds of numbers that cannot be used in measuring. So I think it is preferable to use directly the expression “or a generalization of this concept”.--147.83.48.87 (talk) 18:17, 2 January 2008 (UTC)

I really wish people would drop this useless effort to find a general definition of "number" in the context of this article. You're not going to find it, because there's no such thing -- "number" is a word applied by divers sources to differing collections of concepts, with some commonalities, but no clear demarcation between what belongs and what doesn't. What the article needs to do is simply present the various things that some reasonable fraction of the literature takes to be "numbers", without trying to make them tidier than they actually are. Even the current first sentence overreaches in that direction (how, for example, are octonions "used in counting and measuring"?).

I think the lead sentence should not start a number is... at all, because that formula almost promises that we're going to give a definition, and we can't. A better lead might begin something like

In mathematics, the notion of number is used in various ways, including abstractions used to count objects and measure quantities

Needs polishing, but you can see where I'm going -- we should point in the direction of the most used senses of the word, but without straining to extract a commonality that may not be there, and most especially without giving any warrant to claim that we're excluding anything in particular from numberhood. --Trovatore (talk) 05:39, 3 January 2008 (UTC) "I am not a number, I am a free man!" Rick Norwood (talk) 13:28, 17 January 2008 (UTC)

vid

Wouldn't this work as a definition?

In mathematics, a number system is an algebraic structure consisting of abstract entities called numbers. Number systems typically come with a notion of "size of numbers" defined by an order relation or a norm. Willow1729 (talk) 00:53, 22 September 2008 (UTC)

This doesn't define a number does it? It calls the elements of some set "numbers", but that's a bit self-referential isn't it? I don't think we can ever get a nice precise axiomatic definition down for the concept either - consider that whatever definition we come up with has to deal with systems that have {1, 2, many} as their set of numbers, and other oddities, including systems that aren't closed under any operation. I think at best we can give some abstract description, and then we can give precise examples later. Just my gut feeling on the matter though, I don't have any sources to back it up, or that disagree with you. Cheers, Ben (talk) 11:30, 23 September 2008 (UTC)

I don't think we need a definition of Number, indeed any such definition would be OR. We can list those things which are typically called numbers N, Z,, Q, R, C and a few others. Common properties of these are better left to those properties of algebraic structures they share, which can be stated precisely. --Salix alba (talk) 14:36, 23 September 2008 (UTC)

I know there are lots of difficulties in defining number systems, but I still think it's feasible. I looked through several Wikipedia articles on numbers, and it doesn't look too hard to characterize them mathematically.

First of all, numbers are things that you can manipulate algebraically. It's true that there are sets of numbers that aren't closed under multiplication or addition; for example, the irrational numbers are not closed under these operations. So, if you want to call the irrational numbers a "number system" I guess you should define a number system to be an algebraic structure as I described above or any subset of such a structure.

Second, I don't think you have to worry about things like {1, 2, many} because these don't have articles on Wikipedia. The purpose of defining a number is to provide an abstract characterization of things like N, Z, Q, R, etc.

The point I was trying to make with my definition is that numbers are abstract mathematical entities with algebraic structure. Of course a number system isn't just any algebraic structure; numbers are usually characterized by the fact that there is a notion of "size of numbers". This is true of every number system I can think of the except the sedenions. This system does not have a norm, so it doesn't make sense to talk about the size of a sedenion. However, I don't think this presents any real problem. The sedenions are more of an abstract algebraic structure than a number system. Willow1729 (talk) 23:59, 23 September 2008 (UTC)

I think you're sort of missing the point here. The question is not whether you can come up with a good definition — let's assume for the sake of argument that you can. You still can't put it here, because it would be original research. Even if you can find a sourced definition in the literature, and even if it's a good one, you still can't use it in the opening paragraph and say it's what a number is, because it's not the standard definition in the literature (that's obvious, because there isn't one). (In the second case it might be OK to mention it later in the article.) --Trovatore (talk) 00:30, 24 September 2008 (UTC)

Rational/irrational

Surely the picture next to the real numbers is wrong, as any real number must be either rational or irrational, and the little Venn diagram shows the rationals and irrationals as disjoint subsets of the reals when one should be the complement of the other?

i.e. the label for "irrational" should go in the whole space that's in the reals but not in the rationals, NOT have its own little circle inside the reals (leaving a large section of the reals that is apparently neither rational nor irrational). —Preceding unsigned comment added by 81.159.20.37 (talk) 23:54, 7 April 2009 (UTC)

Agreed, I came here to make the same comment 202.36.179.66 (talk) 07:14, 3 July 2009 (UTC)

I removed the incorrect image. It had been up there for long enough. It is not only wrong because it implies there are real numbers that are neither rational or irrational, it also could mislead one about the relative "sizes" of the sets (i.e. irrational numbers are far more numerous than rational ones, etc.) —Preceding unsigned comment added by 202.36.179.66 (talk) 02:13, 20 July 2009 (UTC)

Definition revisited

An editor has been adding:

A number is the symbolic representation or "name" of a quantity.

I don't think this is exactly correct. A number is abstract, not symbolic. Nor does it even apply to complex numbers. — Arthur Rubin (talk) 17:41, 5 June 2009 (UTC)

Hi Arthur. I think a number is the notation or writting of a quantity. At least, I think such notion should be reflected on the article as a secondary or complementary definition, if you prefer this so. Another possibility, of course, is not to include anything at all about the idea of number as quantity notation, but in this case we wouldn't have a good article. I know in Maths there are many abstractions and generalizations to consider (such as complex numbers, as you mention) that can make the question complex, specially concerning terminological uses. However the concept of number as the notational representation of quantity is the best description I can find according to Mathematics, and I honestly find it at least recommendable to reflect this idea on the article. Bests.

(Also, check out Quantity talk page)--Faustnh (talk) 18:57, 5 June 2009 (UTC)

Technically, from the point of view of mathematical logic, the symbolic representation is not a number, but a numeral. (This is a slight generalization of the term numeral from the everyday use, which is more or less synonymous with digit.)

But the main important point here is that the article absolutely must not attempt to define the notion of "number", because there simply is no single accepted definition. We've been over this bunches of times. No "definition" of number will ever be acceptable by WP standards; it will always be original research, undue weight, or some combination. --Trovatore (talk) 20:10, 5 June 2009 (UTC)

(ec)

I think, perhaps, you (Faustnh) are interpreting a number as represented in a numeral system or as a number name. For instance, the number twelve can be written as "12", "C"x, "1100₂", etc. I don't think any of those is the "number" (although that's a separate philosphical dispute). In my opinion, all of those strings are names of the same number.

Your preferred definition also has problems with real numbers, and really has problems with complex numbers, as there's no order.

— Arthur Rubin (talk) 20:25, 5 June 2009 (UTC)

I'd rather say all of those strings are names of the same quantity.

Anyway, I suggest including in the article the definition of number, as "symbolic representation of quantity", as a restricted primary definition (or a particular, non general, definition). If you still estimate it's not acceptable, then I leave it to your consideration. Bests. --Faustnh (talk) 20:36, 5 June 2009 (UTC)

(PD: in the case of complex numbers, I'd rather say there's no uni-linear order, or I'd even say there isn't still a satisfactory proof about the absolute possibility of an uni-linear ordering). --Faustnh (talk) 20:44, 5 June 2009 (UTC)

I do not believe there is any general definition of "number" in mathematics that encompasses the traditional number systems (N,Z,Q,R,C), ordinal and cardinal numbers, hyperreal and surreal numbers, quaternions and octonians, and all other things that are called numbers. To define a number as a "quantity" seems circular. However, a number is certainly not a representation; that is a numeral. — Carl (CBM · talk) 04:27, 6 June 2009 (UTC)

figure

are there numbers that are real but not rational or irrational? 79.101.174.192 (talk) 17:55, 20 June 2009 (UTC)

ups! complex numbers, of course! :)79.101.174.192 (talk) 17:56, 20 June 2009 (UTC)

ups again! complex are not real, so my above question remains! :)) 79.101.174.192 (talk) 17:57, 20 June 2009 (UTC)

i guess the number may be transcendental numbers, but those are missing in the figure. or not? 79.101.174.192 (talk) 17:59, 20 June 2009 (UTC)

The answer to your first question is "no". If a real number is not rational then, by definition, it is irrational. The diagram is easily misread and somewhat misleading - see the thread "Rational/irrational" above. As for the relationship between transcendental numbers and irrational numbers:

1. A rational number a/b is a solution of the equation bx = a.
2. Therefore all rational numbers are algebraic numbers.
3. Therefore all real transcendental numbers are irrational numbers. Gandalf61 (talk) 16:53, 21 June 2009 (UTC)

thanks! that thread indeed answers my confusion! plus, figure can be fixed by putting 'irrational' in complement part, and 'transcendental' where currently irrational is. 79.101.174.192 (talk) 18:22, 21 June 2009 (UTC)

Removal

The real numbers contain the irrational, rational, integers and natural numbers, and transcendental numbers.

The figure "File:REAL NUMBERS.svg|thumb|300 px|upright|The real numbers contain the irrational, rational, integers and natural numbers, and transcendental numbers." has been removed by 202.36.179.66 [1], on the grounds that it suggests that: The image implies there are real numbers that are neither rational or irrational. Could also mislead one about the relative "sizes" of these sets.

I agree that the figure risks misleading, although it comes down to how one interprets the figure; the caption gives no clue, such as saying that it is a Venn diagram. — Charles Stewart (talk) 07:57, 20 July 2009 (UTC)

History

"speculated that the first known" Did you really write that? —Preceding unsigned comment added by 64.85.211.7 (talk) 18:55, 11 August 2009 (UTC)

GA Review

This review is transcluded from Talk:Number/GA1. The edit link for this section can be used to add comments to the review.

I come to this as an intelligent but ignorant reader, and it is my habit to comment on the article as I read it the first time from printed copy.

I'll thus expand this review over time. It may take a few days to review it fully. I'll make what I consider minor, uncontroversial copy edits, but feel free to revert them. Other suggestions for copy edits I'll list here.

Reviewer: Si Trew (talk) 11:38, 19 July 2010 (UTC)

GA review (see here for criteria)

1. It is reasonably well written.

   a (prose): b (MoS):

   I see no comments on the Talk page about making particular exceptions to MoS typography here, so am going with MoS (particularly WP:MOSNUM) unless it is absurd to do so. For example, minus signs should use −, and fractions should be "of the fraction form" (although it does not define what that is, suggesting only that {{frac}} is available).

   I've made a number of changes for MOS compliance, see the edit summaries.

2. It is factually accurate and verifiable.

   a (references): b (citations to reliable sources): (OR):

   Very few inline citations. I realise this is a general-purpose article and not a deep mathematical article, but when calling out particular theorems or particular mathematicians, it should be referenced better; I've marked a couple of things in particular as {{cn}}, but I think that really every time a formula etc is attributed or a year of discovery mentioned, there should be an inline reference.

   I also note that in the later parts of the history we suddenly start getting date-style inline references (1790) for example, yet no reference for what that refers to.

3. It is broad in its coverage.

   a (major aspects): b (focused):

   Pleasantly surprised that it covers the basics well without going into too much detail.

4. It follows the neutral point of view policy.

   Fair representation without bias:

   The occasional use of flattering or subjective language ("well-known" etc) which I have removed.

5. It is stable.

   No edit wars, etc.:

   Looks fine there.

6. It is illustrated by images, where possible and appropriate.

   a (images are tagged and non-free images have fair use rationales): b (appropriate use with suitable captions):

   There's not a single image in the article. I think it could do with some; the articles on rational numbers, complex numbers and so on have images.

7. Overall:

   Pass/Fail:

   Holding pending better references and a few images, please. Si Trew (talk) 13:53, 19 July 2010 (UTC)

   Unfortunately I feel I have to fail this. There's been no attempt as far as I can see to address my concerns as to citations or use of images. I would have happily given examples of what kind of images to use if there was any attempt to address them, but there has not been. Sadly, I fail on that point.

   If I am mistaken in my view that this article is intended not for the general reader but for mathematicians, this should go to WP:GAR.

   It's a pity since except the lack of images, and the references to particular laws etc which I think could have been easily fixed, this should have been an easy GA pass. I fail it with reluctance. Si Trew (talk) 16:40, 26 July 2010 (UTC)

Copy Edit

Classification of numbers

• The table was headed "Numbers", after talking about "Number systems", which I've changed it to.
• The natural numbers are "one, two, three", then they are "0, 1, 2, 3"..., I don't want this to get too technical right here, but it's an important distinction to make early on whether the natural numbers includes zero or not (i.e. it depends.)

Done. Gandalf61 (talk) 13:01, 19 July 2010 (UTC)

Unfortunately this table has been changed in [this edit] to be far more complex. I had assumed the purpose of this article was as an introduction to the concept of number for people who are not primarily mathematicians (other articles it rightly links to go into more detail); this to me just confuses things. For one thing, the para immediately above it calls them "number systems" and the whole point then is to show examples of number systems, not "counting systems" as this table now has it.

I haven't been doing GA reviews for long so please forgive me if I am out of order. I prefer to pass articles than fail them, and will work with the editors of the articles to achieve that. I just think that edit makes the article worse not better, to the point it would fail my GA review for being too obscure for the nature of the article (i.e. not focused). Si Trew (talk) 19:09, 21 July 2010 (UTC)

Fixed with this change. (ES: Undo good faith edit, far too large with numerous errors.)

I've left I hope a courteous note on the undone editor's talk page, and a welcome. Si Trew (talk) 20:32, 21 July 2010 (UTC)

I think we could remove that whole table. It adds nothing to the explanations in the following paragraphs, and has potential to confuse the reader. Gandalf61 (talk) 09:10, 22 July 2010 (UTC)

That's up to you, I can see the point of it as kind of an intro, and can also see the point of removing it. Either way is fine by me from the point of view of the GA review. Si Trew (talk) 13:18, 22 July 2010 (UTC)

Real numbers

• The sentence starting "In abstract algebra, the real numbers are up to isomorphism uniquely characterized", this is confusing to me (I think may be a slip here).

Reworded. Gandalf61 (talk) 13:01, 19 July 2010 (UTC)

{{frac}}

• WP:MOSMATH#Fractions strongly suggests that {{frac}} not be used. I've converted to {{frac}} to a new template {{fracText}} which meets with the textual form suggested there. If someone wants to restore {{frac}}, it should probably be discussed at the various MOSs. — Arthur Rubin (talk) 22:42, 19 July 2010 (UTC)

That's fine; I'd already commented about the vagueness at WT:MOSNUM#Fractions, but no contributions from other editors there yet. Si Trew (talk) 19:11, 21 July 2010 (UTC)

Not subsets

Strictly speaking the following statement under complex numbers is wrong:

Each of the number systems mentioned above is a proper subset of the next number system. Symbolically, $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}$.

for instance natural numbers are defined by sets containing sets, integers by pairs of naturanl numbers with an equivalence relation under subtraction, rational numbers by a pair of integers again, real numbers by a pair of sets defining a Dedekind cut, or perhaps other definitions but they are not subsets in the set sense.

However we all know what it means, has anyone got a wording that would be more accurate without being too pedantic and silly? Dmcq (talk) 12:43, 19 July 2010 (UTC)

The statement as it stands reflects major reference books. Your objection is technically correct, but we say "A is a subset of B" even when what we really mean is "There is a subset of B isomorphic to A." The isomorphism is understood. This situation is similar to saying "1 + 1 = 2" instead of saying "the number represented by the numeral 1 added to the number represented by the numeral 1 equals the number represented by the numeral 2." Rick Norwood (talk) 13:01, 19 July 2010 (UTC)

Actually, by "A is a subset of B" we mean that "The canonical map from A to B is 1-1,", rather than "There is a subset of B isomorphic to A." Still, the mathematical convention should stand. — Arthur Rubin (talk) 22:38, 19 July 2010 (UTC)

It is a point that bothers me slightly, because I do think the reals are different enough in kind from the simpler structures to justify thinking of, say, the real number zero, as a different object from the natural number zero, and this at a more fundamental level than, say, the choice of a coding via Cauchy sequences or Dedekind cuts. But I doubt there's any useful way to talk about that at the level of an article like number. --Trovatore (talk) 22:48, 19 July 2010 (UTC)

I think it would be counterproductive at number – perhaps I am mistaken, but it should be a general introduction. That is to say, almost everyone knows (or rather thinks they know) what a number is and so this article shows it is a bit more complicated than that, but it doesn't have to dot every I or cross every T, referring to other articles is more appropriate. It should, of course, be correct. Si Trew (talk) 20:59, 30 July 2010 (UTC)

I think the statement above, "we all know what it means", indicates the problem in this article about trying to decide what audience it is aiming at. Si Trew (talk) 21:01, 30 July 2010 (UTC)

Si Trew didn't say, "we all know what it means", he said, "almost everyone knows (or rather thinks they know) what a number is". It is more complicated than most people realize. For example, for centuries major mathematicians denied that a negative number was really a number. This article gives a good working definition and then mentions some of the deeper questions. Rick Norwood (talk) 12:07, 31 July 2010 (UTC)

I would like to mention two points that did not come out clearly enough in the discussion above. 1. numbers are arguably more "primitive" objects than set theory. Thus, the construction of the integers in terms of pairs of natural numbers is just that: a particular construction, which does not change the fact that natural numbers are found among the integers. 2. While there is not much argument about the status of natural, integer, and rational numbers, there is a bit of a controversy about what constitutes the reals. The intuitionist perspective is quite different from the classical one. The current state of the page does not reflect this at all. Perhaps this is appropriate for the level the page is aiming at, or perhaps at least a brief mention should be included. Tkuvho (talk) 10:08, 1 August 2010 (UTC)

I think a discussion of the difference between the intuitionists and the classical perspective would be more appropriate in the article real number. Rick Norwood (talk) 12:49, 1 August 2010 (UTC)

Negative numbers

The article deals with negative numbers in a light and naïve way. It starts talking about negative numbers within the "Integers" section, but the definition ("numbers that are less than zero") would certainly include, among infinitely many others, -1/3, -sqrt(2), and -π. Moreover, the example provided (money in a bank) is also wrong: I can withdraw $123.45 from a bank, and the number representing that withdrawal (-123.45) is certainly not an integer. The problem is not solved by just moving it out of the "Integers section" because the original intention was clearly to deal with negative integers (e.g. "when the set of negative numbers is combined with the natural numbers and zero, the result is the set of integer numbers"). In the context of the hierarchical construction of number sets, when we only have the natural numbers, it makes sense to define the negative integers, and with them and the natural numbers (including 0) get the integers. Perhaps we should try to avoid using the expresion "negative integers" since we have not defined the integers yet (we need the negative integers to do that), but saying just "negative numbers" does not help, because that expresion means a different thing (all of the negative numbers). We could talk about "the numbers which are opposite to the natural numbers" or something along those lines. And clearly, the bank example has no place in the article. El Changuito (talk) 18:58, 22 September 2010 (UTC)

I don't think that it matters that the section implicitly talks about negatives of numbers other than of the integers. You have a good point though, which I think we can indeed simply fix by replacing

"When the set of negative numbers is combined with the natural numbers and zero..."

with

"When the set of the numbers which are opposite to the natural numbers is combined with the natural numbers and zero..."

DVdm (talk) 19:30, 22 September 2010 (UTC)

That will just lead to confusing questions about what 'opposite' means. Negative integers is fine, or negative whole numbers if you really can't bear to use a name before it is defined. Dmcq (talk) 20:04, 22 September 2010 (UTC)

At the article's level of simplicy "opposite to" is just (sort of very loosely) defined. The (subtle, but pertinent) point was that the set of "negative numbers" is too large to be combined with the natural numbers. It's only the "set of negatives of the natural numbers that should be combined with the natural numbers and zero, to produce, by definition, the set of integers. As the phrase stands now, it is just plain wrong. And of course, we can't talk about negative integers before we have introduced integers, which is what we are doing just now. DVdm (talk) 20:41, 22 September 2010 (UTC)

O perfectly well understand what you are saying about the numbers. What I said and what I again repeat is that sticking in another term like opposite of does not help, it is just replacing one word 'negative' with another that isn't yet defined and which is unnecessary and confusing. And there is nothing wrong with using a term before it is formally defined and can be far better than getting oneself tied up trying to avoid the obvious. I suggested using negative whole number as a well understood term which while not defined here will get by for the moment. Dmcq (talk) 20:59, 22 September 2010 (UTC)

You mean like changing the bad phrase to

"When the set of the negative whole numbers is combined with the natural numbers and zero..." ?

I guess that would solve the original problem just as well indeed.

But perhaps we really should informally (but i.m.o. much more properly) first define the negative (or the opposite) of a natural number as something that produces zero when added to the number. Combining the set of these with the set of naturals and zero then produces the integers. DVdm (talk) 21:36, 22 September 2010 (UTC)

I went ahead and made some trivial changes. Feel free to improve. DVdm (talk) 19:36, 23 September 2010 (UTC)

Article Natural_number says zero may or may not be included in set of natural numbers.

I saw an edit that suggested it's unnecessary in this article to mention zero in distinction from natural numbers, because zero is a natural number. But that does not seem to be an invariant property of the term "natural number" in the several sources I have at hand. Many sources start the set of natural numbers at 1, and I think that this article should reflect that uncertainty of definition, as the article Natural_number does. -- WeijiBaikeBianji (talk) 04:14, 24 September 2010 (UTC)

Yes, I agree with you. I looked at that issue briefly a while back but the correct fix wasn't immediately obvious. Shouldn't be too hard to find reasonable language; go for it. --Trovatore (talk) 07:41, 24 September 2010 (UTC)

The current article already reflects this in the section Number#Natural numbers: "Traditionally, the sequence of natural numbers started with 1. However, in the 19th century, set theorists and other mathematicians started the convention of including 0 in the set of natural numbers". DVdm (talk) 14:36, 24 September 2010 (UTC)

The article did not reflect what User DVdm indicates it did. The use of the word "convention" means that it is generally agreed upon to include zero. If I take my books off the shelf, about half will include zero, and about half will not. I fixed this yesterday. The article now indicates how "Natural" may be used to describe two different sets. If you can word it better, feel free of course. Cliff (talk) 16:49, 28 March 2011 (UTC)

Automated theorem proving tends to use the definition with them starting at 1 so we might be heading back to the old ways again! 20:28, 28 March 2011 (UTC)

Irrational Section.

Currently, irrational numbers are only defined in an effort to explain Real numbers. Shouldn't we have a section titled "Irrational numbers" immediately following the section on Rational numbers? It would make more sense to try to explain what an irrational number is in it's own section rather than when trying to define Real numbers. Cliff (talk) 18:04, 14 February 2011 (UTC)

They are described in their own section. There is nothing about the description which says they were only defined in an effort to explain real numbers. Dmcq (talk) 22:13, 14 February 2011 (UTC)

The table of contents reads.

   * 1.1 Natural numbers
   * 1.2 Integers
   * 1.3 Rational numbers
   * 1.4 Real numbers
   * 1.5 Complex numbers
   * 1.6 Computable numbers
   * 1.7 Other types
   * 1.8 Specific uses

This doesn't have Irrational numbers. Is there a reason it doesn't? if not, I'll create the new section.Cliff (talk) 23:44, 14 February 2011 (UTC)

I think this is not a good idea. Irrationals are defined by exclusion — they are simply real numbers that aren't rational. They are not a new "type". --Trovatore (talk) 23:49, 14 February 2011 (UTC)

So you define real numbers before defining irrational numbers? How does that work?Cliff (talk) 00:04, 15 February 2011 (UTC)

Oh, various ways. Take a look at real number for possibly more than you want to know. Let me turn the question around on you: How exactly would you define irrational numbers, without first having defined the real numbers? --Trovatore (talk) 00:09, 15 February 2011 (UTC)

Sorry I looked at the history section where irrationals are quite properly described before going on to the reals as in the discovery of irrational numbers. However nowadays one would as Trovatore says define the reals first and then show there are reals which aren't rational. Dmcq (talk) 10:27, 15 February 2011 (UTC)

Ok. I see what you're saying. My question now is this: What is the point of this article? Is it to be a rigorous mathematics textbook? If so, then we're on the right path. But, if it is to be a reference for someone who might want to know more about numbers, but doesn't have a mathematics background (an encyclopedia), then perhaps we should consider talking about irrational numbers in a way that is approachable to the mainstream. Consider the person who would put "number" into the search bar. Do mathematicians do this (apparently you and I did)? What is our purpose in coming here, and does it fit with the "lay" person's reason for coming here?Cliff (talk) 18:49, 15 February 2011 (UTC)

It is an encyclopaedia rather than a textbook. But besides that I really do think the modern order is easiest. The Greeks didn't have decimal places but now it's the way numbers are expressed even in junior schools. Dmcq (talk) 19:23, 15 February 2011 (UTC)

What do you mean when you say "modern order"Cliff (talk) 19:50, 15 February 2011 (UTC)

I just mean doing the real numbers rather than following the history and dealing with irrationals before getting to the real numbers. Children learn about expressing length using decimal numbers rather than as fractions and the proof of irrationality of the square root of two wouldn't be dealt with in a junior school. Dmcq (talk) 23:32, 15 February 2011 (UTC)

Still not sure what you mean. Do you support the addition of a section on Irrational numbers and how to identify them? Or do you support the status quo? Cliff (talk) 16:51, 28 March 2011 (UTC)

Transcendental numbers and reals

Is there a reason that the histories of these two different sets of numbers are discussed in the same section?Cliff (talk) 00:06, 15 February 2011 (UTC)

Table in classification section

The first three in the table describe sets of numbers, but the last three describe individual elements from the respective sets. If anyone can think about how to fix this, please do. I'll think about it. Cliff (talk) 20:34, 3 April 2011 (UTC)

Describing a general element from a set is one of the standard ways of describing a set see 'special sets' in set (mathematics).Dmcq (talk) 20:45, 3 April 2011 (UTC)

I'm not expressing confusion about what is going on, I'm expressing concern over the clarity of this article as it pertains to the "average" reader, not editor. Cliff (talk) 04:17, 4 April 2011 (UTC)

intro is a mess.

The first sentence is unwieldy and bordering on nonsensical, not a good start to the article. It currently reads: "A number is a mathematical object used to quantify (count and measure) and to represent quantity, in several forms, of which the most primitive, primary and simplest one is the symbolic signification of a particular, invariable, constant quantity." Cliff (talk) 12:43, 20 April 2011 (UTC)

I agree. I have reverted to the previous version, before recent changes by Faus (talk · contribs), which reads "A number is a mathematical object used to count and measure" - much simpler and clearer. Gandalf61 (talk) 13:30, 20 April 2011 (UTC)

...and more inexact. A number is not a mathematical object used to count and measure, but it's rather a mathematical object in a relation of a representational kind with quantity. So you are deleting the wrong clause of the definition, (I left untouched the first, wrong part of the opening definition for not upsetting anyone), but hey, I'm not going to waste my time in permanently monitoring this article, so don't worry, you win. --Faus (talk) 20:43, 21 April 2011 (UTC)

I believe what is there is far better than the proposed change to 'a relation of a representational kind'. Dmcq (talk) 22:10, 21 April 2011 (UTC)

However, what you believe (or I believe) is not the important thing; the important thing is the correct definition of number. But wait, this is Wikipedia... ok then... --Faus (talk) 09:58, 25 April 2011 (UTC)

Faus, an encyclopedia is not a resource for mathematical definitions, but as a resource for people who want to find things out. Imagine a person who might search for the word "number" in an encyclopedia. It is less likely that they're looking for a mathematical definition, but a more general one that will help them understand and not cause further confusion. We're not saying that there is no room in the article for the "correct" definition, but that the intro is not the place for it. Cliff (talk) 20:57, 25 April 2011 (UTC)

Well if it is so much more accurate despite sounding so abstruse where please would I find a source corresponding to a definition of a number as 'a relation of a representational kind'? Dmcq (talk) 22:07, 25 April 2011 (UTC)

As always. 23:58, 25 April 2011 (UTC) — Preceding unsigned comment added by Cliff (talk • contribs)

Lead

Is arithmatic a British spelling Or just a wrong one? Cliff (talk) 17:43, 15 June 2011 (UTC)

It was a spelling mistake introduced by Cpiral (talk · contribs) on June 13. I have reverted to the previous version of the lead, before Cpiral's changes on June 11 and June 13. This previous version was both shorter and clearer than its replacement. Gandalf61 (talk) 08:55, 16 June 2011 (UTC)

Restoring the previous version seems best at this point. I did like the inclusion of talk about how the definition of number is changing with new research. Cliff (talk) 21:13, 16 June 2011 (UTC)

Restore the lead

The new lead, with paragraphs for definition, linguistics, and operations was an improvement in the following ways:

• Puts the multiple ideas pertaining to linguistics in their own paragraph, out of the definition that is the article's actual main content
• Mentions the "reals" with the other basic number classifications mentioned
• Moves "arithmetic" from the paragraph on operations to the paragraph on definition
• Introduces the idea of a digit and positional notation (in the paragraph on linguistics)

It was mostly a copy edit but the whole time I was hoping to place subtle hints about number theory, because some numbers are not used to count and measure. They serve the theory of numbers themselves. To do this I added to the first, defining paragraph the ideas of "operational definition" and the continuing development of number systems, and of number theory (also known as "arithmetic"). I added a link to the theory of number systems (which is what the article is mostly about). I rewrote a sentence that mixed the ideas of "number types" ("negatives" and "zero") and "number systems". Minor fixes: "label" is not the right word; "serial number" and "phone number" are probably overlinked. Cheers. — CpiralCpiral 02:23, 17 June 2011 (UTC)

Cpiral, since your edit I had been thinking about how to fix your work while keeping some of the ideas. I think the main problem was that you had done too much, so were unable to ensure the quality of writing expected. If you'd like to slow down and make smaller improvements that can be edited by others, we can build a better lead as a group. Or we can do it here. I'd like to help, but your lead was confusing rather than enlightening. Cliff (talk) 03:48, 17 June 2011 (UTC)

I did too much. Parts or links did not flow. Thank you.

• "the linguistic use of numbers as word-like or letter-like symbols"
• "...what numbers can be, partly by research and development in pure mathematics."

The kicker is the footnote problem. It made it seem like to much. It was unclear and confusing why such a thing might be said there at the beginning. See next section "intro review". — CpiralCpiral 17:38, 17 June 2011 (UTC)

Review of a new intro

A number is a mathematical object applied in counting and measurement. The history of mathematics provides us an "operational definition" of number, for number's properties are still expanding. A number system is well defined set of numbers and their arithmatic.

Negative numbers appeared around the year zero. Nine hundred years later zero was officially a number used in formal calculations. The fundamental number systems are the naturals (the numbers we count with), the integers (whole numbers), the rationals (fractions), the reals, and the complex numbers. Number theorists are still working to understand what a number can be. Number systems and their histories are summarized here.

The mathematical [[mathematical notation|mathematical notation] for representing numbers is the numeral system. This numeral system extends into linguistics where numbers serve as word-like or letter-like symbols, such as a telephone number or a serial number, or other codes such as an ISBN. The word number has several uses: abstract object, symbol, or word; but this article concerns itself with the numbers that use digits to represent mathematical objects in a positional notation, unlike the "numerals" in linguistics.

Each of the number systems described below defines its numeric operations. The most basic operations are the unary operation, which inputs a single number and outputs a single number, and the binary operation, which inputs two numbers and produces a single output number. The operations of addition, subtraction, multiplication, division, and exponentiation are all binary; but the integer operation "successor" produces, singly, "number plus one". The successor of -10 is -9.

This is just dreadful. I don't think I'll boither contributing to trying to rescue it. I'll simply oppose it. Dmcq (talk) 20:03, 17 June 2011 (UTC)

Agree with Dmcq. There is too much in that proposed intro that is either ungrammatical, poorly worded, unclear or just plain wrong. Oppose. Gandalf61 (talk) 10:03, 18 June 2011 (UTC)